### Speed of sound, Or, No, it’s not freezing in here

I was thinking about my frequency measurements and how I kept getting best fit slopes corresponding to very low values of the speed of sound. I’d hypothesized this might have something to do with the mouthpieces I was using, and I found some possible corroboration in Arthur H. Benade’s book Horns, Strings, and Harmony: “To be sure, an enlarged mouthpiece cavity can lower the pitch of the vibrational modes, but they are lowered equally, except for the highest ones.” A little too vague to make it clear whether the effect looks like a lowered slope, though.

Edit: But yes, it has to. The enlarged cavity should matter less for a long tube than for a short one (since in the latter case it’s a relatively large deviation from a perfect cylinder, while in the former case it’s relatively small.) That means $f$ will be lowered by more for a short tube than for a long one, or in other words the variation in $f$ with $L_{eff}$ will be less than expected; equivalently, the variation in $L_{eff}$ with $1/f$ will be less than expected — that is, the slope will correspond to a lower speed of sound.

Then I realized these tubes are just about the same diameter (inside) as an earbud.

So I shoved an earbud in one end and listened to the other while using this site to play a calibrated variable frequency tone. I could indeed hear the resonances. I couldn’t pin down resonant frequencies with precision better than several hertz, I think, but using several tubes and tube combinations I got enough points for a reasonably accurate fit to a line. After discarding one blatant outlier (yeah, I could be more rigorous, but really, it was an outlier) I got a sound speed at around 19°C of 340.9 m/s — it should be about 342.7. Much closer agreement. I also got an effective length for the “mouthpiece” of –3.5 mm. Negative? Sure, because the earbud was pushed into the tube. By more than 3.5 mm, of course, but I’m sure the error bars on that value are large enough to cover that.

Granted, I did this in the slightly warmer bedroom rather than the chillier basement, but there’s no way the low sound speeds I’d been getting were entirely temperature related. It’s not below freezing down there! So I think it really has to be a mouthpiece effect.